/* ecc.c - TinyCrypt implementation of common ECC functions */

/*
 * Copyright (c) 2014, Kenneth MacKay
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions are met:
 * * Redistributions of source code must retain the above copyright notice,
 * this list of conditions and the following disclaimer.
 * * Redistributions in binary form must reproduce the above copyright notice,
 * this list of conditions and the following disclaimer in the documentation
 * and/or other materials provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
 * WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
 * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR
 * ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
 * (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON
 * ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
 * SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 *
 *  Copyright (C) 2017 by Intel Corporation, All Rights Reserved.
 *
 *  Redistribution and use in source and binary forms, with or without
 *  modification, are permitted provided that the following conditions are met:
 *
 *    - Redistributions of source code must retain the above copyright notice,
 *     this list of conditions and the following disclaimer.
 *
 *    - Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 *
 *    - Neither the name of Intel Corporation nor the names of its contributors
 *    may be used to endorse or promote products derived from this software
 *    without specific prior written permission.
 *
 *  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
 *  AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 *  IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 *  ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
 *  LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
 *  CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
 *  SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
 *  INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
 *  CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 *  ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
 *  POSSIBILITY OF SUCH DAMAGE.
 */

#include "ecc.h"
#include "../include/tinycrypt/ecc.h"
#include "ecc_platform_specific.h"
#include <string.h>

/* IMPORTANT: Make sure a cryptographically-secure PRNG is set and the platform
 * has access to enough entropy in order to feed the PRNG regularly. */
#if default_RNG_defined
static uECC_RNG_Function g_rng_function = &default_CSPRNG;
#else
static uECC_RNG_Function g_rng_function = 0;
#endif

void uECC_set_rng(uECC_RNG_Function rng_function) { g_rng_function = rng_function; }

uECC_RNG_Function uECC_get_rng(void) { return g_rng_function; }

int uECC_curve_private_key_size(uECC_Curve curve) { return BITS_TO_BYTES(curve->num_n_bits); }

int uECC_curve_public_key_size(uECC_Curve curve) { return 2 * curve->num_bytes; }

void uECC_vli_clear(uECC_word_t *vli, wordcount_t num_words) {
  wordcount_t i;
  for (i = 0; i < num_words; ++i) {
    vli[i] = 0;
  }
}

uECC_word_t uECC_vli_isZero(const uECC_word_t *vli, wordcount_t num_words) {
  uECC_word_t bits = 0;
  wordcount_t i;
  for (i = 0; i < num_words; ++i) {
    bits |= vli[i];
  }
  return (bits == 0);
}

uECC_word_t uECC_vli_testBit(const uECC_word_t *vli, bitcount_t bit) { return (vli[bit >> uECC_WORD_BITS_SHIFT] & ((uECC_word_t)1 << (bit & uECC_WORD_BITS_MASK))); }

/* Counts the number of words in vli. */
static wordcount_t vli_numDigits(const uECC_word_t *vli, const wordcount_t max_words) {
  wordcount_t i;
  /* Search from the end until we find a non-zero digit. We do it in reverse
   * because we expect that most digits will be nonzero. */
  for (i = max_words - 1; i >= 0 && vli[i] == 0; --i) {
  }

  return (i + 1);
}

bitcount_t uECC_vli_numBits(const uECC_word_t *vli, const wordcount_t max_words) {
  uECC_word_t i;
  uECC_word_t digit;

  wordcount_t num_digits = vli_numDigits(vli, max_words);
  if (num_digits == 0) {
    return 0;
  }

  digit = vli[num_digits - 1];
  for (i = 0; digit; ++i) {
    digit >>= 1;
  }

  return (((bitcount_t)(num_digits - 1) << uECC_WORD_BITS_SHIFT) + i);
}

void uECC_vli_set(uECC_word_t *dest, const uECC_word_t *src, wordcount_t num_words) {
  wordcount_t i;

  for (i = 0; i < num_words; ++i) {
    dest[i] = src[i];
  }
}

cmpresult_t uECC_vli_cmp_unsafe(const uECC_word_t *left, const uECC_word_t *right, wordcount_t num_words) {
  wordcount_t i;

  for (i = num_words - 1; i >= 0; --i) {
    if (left[i] > right[i]) {
      return 1;
    } else if (left[i] < right[i]) {
      return -1;
    }
  }
  return 0;
}

uECC_word_t uECC_vli_equal(const uECC_word_t *left, const uECC_word_t *right, wordcount_t num_words) {
  uECC_word_t diff = 0;
  wordcount_t i;

  for (i = num_words - 1; i >= 0; --i) {
    diff |= (left[i] ^ right[i]);
  }
  return !(diff == 0);
}

uECC_word_t cond_set(uECC_word_t p_true, uECC_word_t p_false, unsigned int cond) { return (p_true * (cond)) | (p_false * (!cond)); }

/* Computes result = left - right, returning borrow, in constant time.
 * Can modify in place. */
uECC_word_t uECC_vli_sub(uECC_word_t *result, const uECC_word_t *left, const uECC_word_t *right, wordcount_t num_words) {
  uECC_word_t borrow = 0;
  wordcount_t i;
  for (i = 0; i < num_words; ++i) {
    uECC_word_t diff = left[i] - right[i] - borrow;
    uECC_word_t val  = (diff > left[i]);
    borrow           = cond_set(val, borrow, (diff != left[i]));

    result[i] = diff;
  }
  return borrow;
}

/* Computes result = left + right, returning carry, in constant time.
 * Can modify in place. */
static uECC_word_t uECC_vli_add(uECC_word_t *result, const uECC_word_t *left, const uECC_word_t *right, wordcount_t num_words) {
  uECC_word_t carry = 0;
  wordcount_t i;
  for (i = 0; i < num_words; ++i) {
    uECC_word_t sum = left[i] + right[i] + carry;
    uECC_word_t val = (sum < left[i]);
    carry           = cond_set(val, carry, (sum != left[i]));
    result[i]       = sum;
  }
  return carry;
}

cmpresult_t uECC_vli_cmp(const uECC_word_t *left, const uECC_word_t *right, wordcount_t num_words) {
  uECC_word_t tmp[NUM_ECC_WORDS];
  uECC_word_t neg   = !!uECC_vli_sub(tmp, left, right, num_words);
  uECC_word_t equal = uECC_vli_isZero(tmp, num_words);
  return (!equal - 2 * neg);
}

/* Computes vli = vli >> 1. */
static void uECC_vli_rshift1(uECC_word_t *vli, wordcount_t num_words) {
  uECC_word_t *end   = vli;
  uECC_word_t  carry = 0;

  vli += num_words;
  while (vli-- > end) {
    uECC_word_t temp = *vli;
    *vli             = (temp >> 1) | carry;
    carry            = temp << (uECC_WORD_BITS - 1);
  }
}

static void muladd(uECC_word_t a, uECC_word_t b, uECC_word_t *r0, uECC_word_t *r1, uECC_word_t *r2) {
  uECC_dword_t p   = (uECC_dword_t)a * b;
  uECC_dword_t r01 = ((uECC_dword_t)(*r1) << uECC_WORD_BITS) | *r0;
  r01 += p;
  *r2 += (r01 < p);
  *r1 = r01 >> uECC_WORD_BITS;
  *r0 = (uECC_word_t)r01;
}

/* Computes result = left * right. Result must be 2 * num_words long. */
static void uECC_vli_mult(uECC_word_t *result, const uECC_word_t *left, const uECC_word_t *right, wordcount_t num_words) {
  uECC_word_t r0 = 0;
  uECC_word_t r1 = 0;
  uECC_word_t r2 = 0;
  wordcount_t i, k;

  /* Compute each digit of result in sequence, maintaining the carries. */
  for (k = 0; k < num_words; ++k) {
    for (i = 0; i <= k; ++i) {
      muladd(left[i], right[k - i], &r0, &r1, &r2);
    }

    result[k] = r0;
    r0        = r1;
    r1        = r2;
    r2        = 0;
  }

  for (k = num_words; k < num_words * 2 - 1; ++k) {
    for (i = (k + 1) - num_words; i < num_words; ++i) {
      muladd(left[i], right[k - i], &r0, &r1, &r2);
    }
    result[k] = r0;
    r0        = r1;
    r1        = r2;
    r2        = 0;
  }
  result[num_words * 2 - 1] = r0;
}

void uECC_vli_modAdd(uECC_word_t *result, const uECC_word_t *left, const uECC_word_t *right, const uECC_word_t *mod, wordcount_t num_words) {
  uECC_word_t carry = uECC_vli_add(result, left, right, num_words);
  if (carry || uECC_vli_cmp_unsafe(mod, result, num_words) != 1) {
    /* result > mod (result = mod + remainder), so subtract mod to get
     * remainder. */
    uECC_vli_sub(result, result, mod, num_words);
  }
}

void uECC_vli_modSub(uECC_word_t *result, const uECC_word_t *left, const uECC_word_t *right, const uECC_word_t *mod, wordcount_t num_words) {
  uECC_word_t l_borrow = uECC_vli_sub(result, left, right, num_words);
  if (l_borrow) {
    /* In this case, result == -diff == (max int) - diff. Since -x % d == d - x,
     * we can get the correct result from result + mod (with overflow). */
    uECC_vli_add(result, result, mod, num_words);
  }
}

/* Computes result = product % mod, where product is 2N words long. */
/* Currently only designed to work for curve_p or curve_n. */
void uECC_vli_mmod(uECC_word_t *result, uECC_word_t *product, const uECC_word_t *mod, wordcount_t num_words) {
  uECC_word_t  mod_multiple[2 * NUM_ECC_WORDS];
  uECC_word_t  tmp[2 * NUM_ECC_WORDS];
  uECC_word_t *v[2] = {tmp, product};
  uECC_word_t  index;

  /* Shift mod so its highest set bit is at the maximum position. */
  bitcount_t  shift      = (num_words * 2 * uECC_WORD_BITS) - uECC_vli_numBits(mod, num_words);
  wordcount_t word_shift = shift / uECC_WORD_BITS;
  wordcount_t bit_shift  = shift % uECC_WORD_BITS;
  uECC_word_t carry      = 0;
  uECC_vli_clear(mod_multiple, word_shift);
  if (bit_shift > 0) {
    for (index = 0; index < (uECC_word_t)num_words; ++index) {
      mod_multiple[word_shift + index] = (mod[index] << bit_shift) | carry;
      carry                            = mod[index] >> (uECC_WORD_BITS - bit_shift);
    }
  } else {
    uECC_vli_set(mod_multiple + word_shift, mod, num_words);
  }

  for (index = 1; shift >= 0; --shift) {
    uECC_word_t borrow = 0;
    wordcount_t i;
    for (i = 0; i < num_words * 2; ++i) {
      uECC_word_t diff = v[index][i] - mod_multiple[i] - borrow;
      if (diff != v[index][i]) {
        borrow = (diff > v[index][i]);
      }
      v[1 - index][i] = diff;
    }
    /* Swap the index if there was no borrow */
    index = !(index ^ borrow);
    uECC_vli_rshift1(mod_multiple, num_words);
    mod_multiple[num_words - 1] |= mod_multiple[num_words] << (uECC_WORD_BITS - 1);
    uECC_vli_rshift1(mod_multiple + num_words, num_words);
  }
  uECC_vli_set(result, v[index], num_words);
}

void uECC_vli_modMult(uECC_word_t *result, const uECC_word_t *left, const uECC_word_t *right, const uECC_word_t *mod, wordcount_t num_words) {
  uECC_word_t product[2 * NUM_ECC_WORDS];
  uECC_vli_mult(product, left, right, num_words);
  uECC_vli_mmod(result, product, mod, num_words);
}

void uECC_vli_modMult_fast(uECC_word_t *result, const uECC_word_t *left, const uECC_word_t *right, uECC_Curve curve) {
  uECC_word_t product[2 * NUM_ECC_WORDS];
  uECC_vli_mult(product, left, right, curve->num_words);

  curve->mmod_fast(result, product);
}

static void uECC_vli_modSquare_fast(uECC_word_t *result, const uECC_word_t *left, uECC_Curve curve) { uECC_vli_modMult_fast(result, left, left, curve); }

#define EVEN(vli) (!(vli[0] & 1))

static void vli_modInv_update(uECC_word_t *uv, const uECC_word_t *mod, wordcount_t num_words) {
  uECC_word_t carry = 0;

  if (!EVEN(uv)) {
    carry = uECC_vli_add(uv, uv, mod, num_words);
  }
  uECC_vli_rshift1(uv, num_words);
  if (carry) {
    uv[num_words - 1] |= HIGH_BIT_SET;
  }
}

void uECC_vli_modInv(uECC_word_t *result, const uECC_word_t *input, const uECC_word_t *mod, wordcount_t num_words) {
  uECC_word_t a[NUM_ECC_WORDS], b[NUM_ECC_WORDS];
  uECC_word_t u[NUM_ECC_WORDS], v[NUM_ECC_WORDS];
  cmpresult_t cmpResult;

  if (uECC_vli_isZero(input, num_words)) {
    uECC_vli_clear(result, num_words);
    return;
  }

  uECC_vli_set(a, input, num_words);
  uECC_vli_set(b, mod, num_words);
  uECC_vli_clear(u, num_words);
  u[0] = 1;
  uECC_vli_clear(v, num_words);
  while ((cmpResult = uECC_vli_cmp_unsafe(a, b, num_words)) != 0) {
    if (EVEN(a)) {
      uECC_vli_rshift1(a, num_words);
      vli_modInv_update(u, mod, num_words);
    } else if (EVEN(b)) {
      uECC_vli_rshift1(b, num_words);
      vli_modInv_update(v, mod, num_words);
    } else if (cmpResult > 0) {
      uECC_vli_sub(a, a, b, num_words);
      uECC_vli_rshift1(a, num_words);
      if (uECC_vli_cmp_unsafe(u, v, num_words) < 0) {
        uECC_vli_add(u, u, mod, num_words);
      }
      uECC_vli_sub(u, u, v, num_words);
      vli_modInv_update(u, mod, num_words);
    } else {
      uECC_vli_sub(b, b, a, num_words);
      uECC_vli_rshift1(b, num_words);
      if (uECC_vli_cmp_unsafe(v, u, num_words) < 0) {
        uECC_vli_add(v, v, mod, num_words);
      }
      uECC_vli_sub(v, v, u, num_words);
      vli_modInv_update(v, mod, num_words);
    }
  }
  uECC_vli_set(result, u, num_words);
}

/* ------ Point operations ------ */

void double_jacobian_default(uECC_word_t *X1, uECC_word_t *Y1, uECC_word_t *Z1, uECC_Curve curve) {
  /* t1 = X, t2 = Y, t3 = Z */
  uECC_word_t t4[NUM_ECC_WORDS];
  uECC_word_t t5[NUM_ECC_WORDS];
  wordcount_t num_words = curve->num_words;

  if (uECC_vli_isZero(Z1, num_words)) {
    return;
  }

  uECC_vli_modSquare_fast(t4, Y1, curve);   /* t4 = y1^2 */
  uECC_vli_modMult_fast(t5, X1, t4, curve); /* t5 = x1*y1^2 = A */
  uECC_vli_modSquare_fast(t4, t4, curve);   /* t4 = y1^4 */
  uECC_vli_modMult_fast(Y1, Y1, Z1, curve); /* t2 = y1*z1 = z3 */
  uECC_vli_modSquare_fast(Z1, Z1, curve);   /* t3 = z1^2 */

  uECC_vli_modAdd(X1, X1, Z1, curve->p, num_words); /* t1 = x1 + z1^2 */
  uECC_vli_modAdd(Z1, Z1, Z1, curve->p, num_words); /* t3 = 2*z1^2 */
  uECC_vli_modSub(Z1, X1, Z1, curve->p, num_words); /* t3 = x1 - z1^2 */
  uECC_vli_modMult_fast(X1, X1, Z1, curve);         /* t1 = x1^2 - z1^4 */

  uECC_vli_modAdd(Z1, X1, X1, curve->p, num_words); /* t3 = 2*(x1^2 - z1^4) */
  uECC_vli_modAdd(X1, X1, Z1, curve->p, num_words); /* t1 = 3*(x1^2 - z1^4) */
  if (uECC_vli_testBit(X1, 0)) {
    uECC_word_t l_carry = uECC_vli_add(X1, X1, curve->p, num_words);
    uECC_vli_rshift1(X1, num_words);
    X1[num_words - 1] |= l_carry << (uECC_WORD_BITS - 1);
  } else {
    uECC_vli_rshift1(X1, num_words);
  }

  /* t1 = 3/2*(x1^2 - z1^4) = B */
  uECC_vli_modSquare_fast(Z1, X1, curve);           /* t3 = B^2 */
  uECC_vli_modSub(Z1, Z1, t5, curve->p, num_words); /* t3 = B^2 - A */
  uECC_vli_modSub(Z1, Z1, t5, curve->p, num_words); /* t3 = B^2 - 2A = x3 */
  uECC_vli_modSub(t5, t5, Z1, curve->p, num_words); /* t5 = A - x3 */
  uECC_vli_modMult_fast(X1, X1, t5, curve);         /* t1 = B * (A - x3) */
  /* t4 = B * (A - x3) - y1^4 = y3: */
  uECC_vli_modSub(t4, X1, t4, curve->p, num_words);

  uECC_vli_set(X1, Z1, num_words);
  uECC_vli_set(Z1, Y1, num_words);
  uECC_vli_set(Y1, t4, num_words);
}

void x_side_default(uECC_word_t *result, const uECC_word_t *x, uECC_Curve curve) {
  uECC_word_t _3[NUM_ECC_WORDS] = {3}; /* -a = 3 */
  wordcount_t num_words         = curve->num_words;

  uECC_vli_modSquare_fast(result, x, curve);                /* r = x^2 */
  uECC_vli_modSub(result, result, _3, curve->p, num_words); /* r = x^2 - 3 */
  uECC_vli_modMult_fast(result, result, x, curve);          /* r = x^3 - 3x */
  /* r = x^3 - 3x + b: */
  uECC_vli_modAdd(result, result, curve->b, curve->p, num_words);
}

uECC_Curve uECC_secp256r1(void) { return &curve_secp256r1; }

void vli_mmod_fast_secp256r1(unsigned int *result, unsigned int *product) {
  unsigned int tmp[NUM_ECC_WORDS];
  int          carry;

  /* t */
  uECC_vli_set(result, product, NUM_ECC_WORDS);

  /* s1 */
  tmp[0] = tmp[1] = tmp[2] = 0;
  tmp[3]                   = product[11];
  tmp[4]                   = product[12];
  tmp[5]                   = product[13];
  tmp[6]                   = product[14];
  tmp[7]                   = product[15];
  carry                    = uECC_vli_add(tmp, tmp, tmp, NUM_ECC_WORDS);
  carry += uECC_vli_add(result, result, tmp, NUM_ECC_WORDS);

  /* s2 */
  tmp[3] = product[12];
  tmp[4] = product[13];
  tmp[5] = product[14];
  tmp[6] = product[15];
  tmp[7] = 0;
  carry += uECC_vli_add(tmp, tmp, tmp, NUM_ECC_WORDS);
  carry += uECC_vli_add(result, result, tmp, NUM_ECC_WORDS);

  /* s3 */
  tmp[0] = product[8];
  tmp[1] = product[9];
  tmp[2] = product[10];
  tmp[3] = tmp[4] = tmp[5] = 0;
  tmp[6]                   = product[14];
  tmp[7]                   = product[15];
  carry += uECC_vli_add(result, result, tmp, NUM_ECC_WORDS);

  /* s4 */
  tmp[0] = product[9];
  tmp[1] = product[10];
  tmp[2] = product[11];
  tmp[3] = product[13];
  tmp[4] = product[14];
  tmp[5] = product[15];
  tmp[6] = product[13];
  tmp[7] = product[8];
  carry += uECC_vli_add(result, result, tmp, NUM_ECC_WORDS);

  /* d1 */
  tmp[0] = product[11];
  tmp[1] = product[12];
  tmp[2] = product[13];
  tmp[3] = tmp[4] = tmp[5] = 0;
  tmp[6]                   = product[8];
  tmp[7]                   = product[10];
  carry -= uECC_vli_sub(result, result, tmp, NUM_ECC_WORDS);

  /* d2 */
  tmp[0] = product[12];
  tmp[1] = product[13];
  tmp[2] = product[14];
  tmp[3] = product[15];
  tmp[4] = tmp[5] = 0;
  tmp[6]          = product[9];
  tmp[7]          = product[11];
  carry -= uECC_vli_sub(result, result, tmp, NUM_ECC_WORDS);

  /* d3 */
  tmp[0] = product[13];
  tmp[1] = product[14];
  tmp[2] = product[15];
  tmp[3] = product[8];
  tmp[4] = product[9];
  tmp[5] = product[10];
  tmp[6] = 0;
  tmp[7] = product[12];
  carry -= uECC_vli_sub(result, result, tmp, NUM_ECC_WORDS);

  /* d4 */
  tmp[0] = product[14];
  tmp[1] = product[15];
  tmp[2] = 0;
  tmp[3] = product[9];
  tmp[4] = product[10];
  tmp[5] = product[11];
  tmp[6] = 0;
  tmp[7] = product[13];
  carry -= uECC_vli_sub(result, result, tmp, NUM_ECC_WORDS);

  if (carry < 0) {
    do {
      carry += uECC_vli_add(result, result, curve_secp256r1.p, NUM_ECC_WORDS);
    } while (carry < 0);
  } else {
    while (carry || uECC_vli_cmp_unsafe(curve_secp256r1.p, result, NUM_ECC_WORDS) != 1) {
      carry -= uECC_vli_sub(result, result, curve_secp256r1.p, NUM_ECC_WORDS);
    }
  }
}

uECC_word_t EccPoint_isZero(const uECC_word_t *point, uECC_Curve curve) { return uECC_vli_isZero(point, curve->num_words * 2); }

void apply_z(uECC_word_t *X1, uECC_word_t *Y1, const uECC_word_t *const Z, uECC_Curve curve) {
  uECC_word_t t1[NUM_ECC_WORDS];

  uECC_vli_modSquare_fast(t1, Z, curve);    /* z^2 */
  uECC_vli_modMult_fast(X1, X1, t1, curve); /* x1 * z^2 */
  uECC_vli_modMult_fast(t1, t1, Z, curve);  /* z^3 */
  uECC_vli_modMult_fast(Y1, Y1, t1, curve); /* y1 * z^3 */
}

/* P = (x1, y1) => 2P, (x2, y2) => P' */
static void XYcZ_initial_double(uECC_word_t *X1, uECC_word_t *Y1, uECC_word_t *X2, uECC_word_t *Y2, const uECC_word_t *const initial_Z, uECC_Curve curve) {
  uECC_word_t z[NUM_ECC_WORDS];
  wordcount_t num_words = curve->num_words;
  if (initial_Z) {
    uECC_vli_set(z, initial_Z, num_words);
  } else {
    uECC_vli_clear(z, num_words);
    z[0] = 1;
  }

  uECC_vli_set(X2, X1, num_words);
  uECC_vli_set(Y2, Y1, num_words);

  apply_z(X1, Y1, z, curve);
  curve->double_jacobian(X1, Y1, z, curve);
  apply_z(X2, Y2, z, curve);
}

void XYcZ_add(uECC_word_t *X1, uECC_word_t *Y1, uECC_word_t *X2, uECC_word_t *Y2, uECC_Curve curve) {
  /* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */
  uECC_word_t t5[NUM_ECC_WORDS];
  wordcount_t num_words = curve->num_words;

  uECC_vli_modSub(t5, X2, X1, curve->p, num_words); /* t5 = x2 - x1 */
  uECC_vli_modSquare_fast(t5, t5, curve);           /* t5 = (x2 - x1)^2 = A */
  uECC_vli_modMult_fast(X1, X1, t5, curve);         /* t1 = x1*A = B */
  uECC_vli_modMult_fast(X2, X2, t5, curve);         /* t3 = x2*A = C */
  uECC_vli_modSub(Y2, Y2, Y1, curve->p, num_words); /* t4 = y2 - y1 */
  uECC_vli_modSquare_fast(t5, Y2, curve);           /* t5 = (y2 - y1)^2 = D */

  uECC_vli_modSub(t5, t5, X1, curve->p, num_words); /* t5 = D - B */
  uECC_vli_modSub(t5, t5, X2, curve->p, num_words); /* t5 = D - B - C = x3 */
  uECC_vli_modSub(X2, X2, X1, curve->p, num_words); /* t3 = C - B */
  uECC_vli_modMult_fast(Y1, Y1, X2, curve);         /* t2 = y1*(C - B) */
  uECC_vli_modSub(X2, X1, t5, curve->p, num_words); /* t3 = B - x3 */
  uECC_vli_modMult_fast(Y2, Y2, X2, curve);         /* t4 = (y2 - y1)*(B - x3) */
  uECC_vli_modSub(Y2, Y2, Y1, curve->p, num_words); /* t4 = y3 */

  uECC_vli_set(X2, t5, num_words);
}

/* Input P = (x1, y1, Z), Q = (x2, y2, Z)
   Output P + Q = (x3, y3, Z3), P - Q = (x3', y3', Z3)
   or P => P - Q, Q => P + Q
 */
static void XYcZ_addC(uECC_word_t *X1, uECC_word_t *Y1, uECC_word_t *X2, uECC_word_t *Y2, uECC_Curve curve) {
  /* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */
  uECC_word_t t5[NUM_ECC_WORDS];
  uECC_word_t t6[NUM_ECC_WORDS];
  uECC_word_t t7[NUM_ECC_WORDS];
  wordcount_t num_words = curve->num_words;

  uECC_vli_modSub(t5, X2, X1, curve->p, num_words); /* t5 = x2 - x1 */
  uECC_vli_modSquare_fast(t5, t5, curve);           /* t5 = (x2 - x1)^2 = A */
  uECC_vli_modMult_fast(X1, X1, t5, curve);         /* t1 = x1*A = B */
  uECC_vli_modMult_fast(X2, X2, t5, curve);         /* t3 = x2*A = C */
  uECC_vli_modAdd(t5, Y2, Y1, curve->p, num_words); /* t5 = y2 + y1 */
  uECC_vli_modSub(Y2, Y2, Y1, curve->p, num_words); /* t4 = y2 - y1 */

  uECC_vli_modSub(t6, X2, X1, curve->p, num_words); /* t6 = C - B */
  uECC_vli_modMult_fast(Y1, Y1, t6, curve);         /* t2 = y1 * (C - B) = E */
  uECC_vli_modAdd(t6, X1, X2, curve->p, num_words); /* t6 = B + C */
  uECC_vli_modSquare_fast(X2, Y2, curve);           /* t3 = (y2 - y1)^2 = D */
  uECC_vli_modSub(X2, X2, t6, curve->p, num_words); /* t3 = D - (B + C) = x3 */

  uECC_vli_modSub(t7, X1, X2, curve->p, num_words); /* t7 = B - x3 */
  uECC_vli_modMult_fast(Y2, Y2, t7, curve);         /* t4 = (y2 - y1)*(B - x3) */
  /* t4 = (y2 - y1)*(B - x3) - E = y3: */
  uECC_vli_modSub(Y2, Y2, Y1, curve->p, num_words);

  uECC_vli_modSquare_fast(t7, t5, curve);           /* t7 = (y2 + y1)^2 = F */
  uECC_vli_modSub(t7, t7, t6, curve->p, num_words); /* t7 = F - (B + C) = x3' */
  uECC_vli_modSub(t6, t7, X1, curve->p, num_words); /* t6 = x3' - B */
  uECC_vli_modMult_fast(t6, t6, t5, curve);         /* t6 = (y2+y1)*(x3' - B) */
  /* t2 = (y2+y1)*(x3' - B) - E = y3': */
  uECC_vli_modSub(Y1, t6, Y1, curve->p, num_words);

  uECC_vli_set(X1, t7, num_words);
}

void EccPoint_mult(uECC_word_t *result, const uECC_word_t *point, const uECC_word_t *scalar, const uECC_word_t *initial_Z, bitcount_t num_bits, uECC_Curve curve) {
  /* R0 and R1 */
  uECC_word_t Rx[2][NUM_ECC_WORDS];
  uECC_word_t Ry[2][NUM_ECC_WORDS];
  uECC_word_t z[NUM_ECC_WORDS];
  bitcount_t  i;
  uECC_word_t nb;
  wordcount_t num_words = curve->num_words;

  uECC_vli_set(Rx[1], point, num_words);
  uECC_vli_set(Ry[1], point + num_words, num_words);

  XYcZ_initial_double(Rx[1], Ry[1], Rx[0], Ry[0], initial_Z, curve);

  for (i = num_bits - 2; i > 0; --i) {
    nb = !uECC_vli_testBit(scalar, i);
    XYcZ_addC(Rx[1 - nb], Ry[1 - nb], Rx[nb], Ry[nb], curve);
    XYcZ_add(Rx[nb], Ry[nb], Rx[1 - nb], Ry[1 - nb], curve);
  }

  nb = !uECC_vli_testBit(scalar, 0);
  XYcZ_addC(Rx[1 - nb], Ry[1 - nb], Rx[nb], Ry[nb], curve);

  /* Find final 1/Z value. */
  uECC_vli_modSub(z, Rx[1], Rx[0], curve->p, num_words); /* X1 - X0 */
  uECC_vli_modMult_fast(z, z, Ry[1 - nb], curve);        /* Yb * (X1 - X0) */
  uECC_vli_modMult_fast(z, z, point, curve);             /* xP * Yb * (X1 - X0) */
  uECC_vli_modInv(z, z, curve->p, num_words);            /* 1 / (xP * Yb * (X1 - X0))*/
  /* yP / (xP * Yb * (X1 - X0)) */
  uECC_vli_modMult_fast(z, z, point + num_words, curve);
  /* Xb * yP / (xP * Yb * (X1 - X0)) */
  uECC_vli_modMult_fast(z, z, Rx[1 - nb], curve);
  /* End 1/Z calculation */

  XYcZ_add(Rx[nb], Ry[nb], Rx[1 - nb], Ry[1 - nb], curve);
  apply_z(Rx[0], Ry[0], z, curve);

  uECC_vli_set(result, Rx[0], num_words);
  uECC_vli_set(result + num_words, Ry[0], num_words);
}

uECC_word_t regularize_k(const uECC_word_t *const k, uECC_word_t *k0, uECC_word_t *k1, uECC_Curve curve) {
  wordcount_t num_n_words = BITS_TO_WORDS(curve->num_n_bits);

  bitcount_t num_n_bits = curve->num_n_bits;

  uECC_word_t carry = uECC_vli_add(k0, k, curve->n, num_n_words) || (num_n_bits < ((bitcount_t)num_n_words * uECC_WORD_SIZE * 8) && uECC_vli_testBit(k0, num_n_bits));

  uECC_vli_add(k1, k0, curve->n, num_n_words);

  return carry;
}

uECC_word_t EccPoint_compute_public_key(uECC_word_t *result, uECC_word_t *private_key, uECC_Curve curve) {
  uECC_word_t  tmp1[NUM_ECC_WORDS];
  uECC_word_t  tmp2[NUM_ECC_WORDS];
  uECC_word_t *p2[2] = {tmp1, tmp2};
  uECC_word_t  carry;

  /* Regularize the bitcount for the private key so that attackers cannot
   * use a side channel attack to learn the number of leading zeros. */
  carry = regularize_k(private_key, tmp1, tmp2, curve);

  EccPoint_mult(result, curve->G, p2[!carry], 0, curve->num_n_bits + 1, curve);

  if (EccPoint_isZero(result, curve)) {
    return 0;
  }
  return 1;
}

/* Converts an integer in uECC native format to big-endian bytes. */
void uECC_vli_nativeToBytes(uint8_t *bytes, int num_bytes, const unsigned int *native) {
  wordcount_t i;
  for (i = 0; i < num_bytes; ++i) {
    unsigned b = num_bytes - 1 - i;
    bytes[i]   = native[b / uECC_WORD_SIZE] >> (8 * (b % uECC_WORD_SIZE));
  }
}

/* Converts big-endian bytes to an integer in uECC native format. */
void uECC_vli_bytesToNative(unsigned int *native, const uint8_t *bytes, int num_bytes) {
  wordcount_t i;
  uECC_vli_clear(native, (num_bytes + (uECC_WORD_SIZE - 1)) / uECC_WORD_SIZE);
  for (i = 0; i < num_bytes; ++i) {
    unsigned b = num_bytes - 1 - i;
    native[b / uECC_WORD_SIZE] |= (uECC_word_t)bytes[i] << (8 * (b % uECC_WORD_SIZE));
  }
}

int uECC_generate_random_int(uECC_word_t *random, const uECC_word_t *top, wordcount_t num_words) {
  uECC_word_t mask = (uECC_word_t)-1;
  uECC_word_t tries;
  bitcount_t  num_bits = uECC_vli_numBits(top, num_words);

  if (!g_rng_function) {
    return 0;
  }

  for (tries = 0; tries < uECC_RNG_MAX_TRIES; ++tries) {
    if (!g_rng_function((uint8_t *)random, num_words * uECC_WORD_SIZE)) {
      return 0;
    }
    random[num_words - 1] &= mask >> ((bitcount_t)(num_words * uECC_WORD_SIZE * 8 - num_bits));
    if (!uECC_vli_isZero(random, num_words) && uECC_vli_cmp(top, random, num_words) == 1) {
      return 1;
    }
  }
  return 0;
}

int uECC_valid_point(const uECC_word_t *point, uECC_Curve curve) {
  uECC_word_t tmp1[NUM_ECC_WORDS];
  uECC_word_t tmp2[NUM_ECC_WORDS];
  wordcount_t num_words = curve->num_words;

  /* The point at infinity is invalid. */
  if (EccPoint_isZero(point, curve)) {
    return -1;
  }

  /* x and y must be smaller than p. */
  if (uECC_vli_cmp_unsafe(curve->p, point, num_words) != 1 || uECC_vli_cmp_unsafe(curve->p, point + num_words, num_words) != 1) {
    return -2;
  }

  uECC_vli_modSquare_fast(tmp1, point + num_words, curve);
  curve->x_side(tmp2, point, curve); /* tmp2 = x^3 + ax + b */

  /* Make sure that y^2 == x^3 + ax + b */
  if (uECC_vli_equal(tmp1, tmp2, num_words) != 0)
    return -3;

  return 0;
}

int uECC_valid_public_key(const uint8_t *public_key, uECC_Curve curve) {
  uECC_word_t _public[NUM_ECC_WORDS * 2];

  uECC_vli_bytesToNative(_public, public_key, curve->num_bytes);
  uECC_vli_bytesToNative(_public + curve->num_words, public_key + curve->num_bytes, curve->num_bytes);

  if (uECC_vli_cmp_unsafe(_public, curve->G, NUM_ECC_WORDS * 2) == 0) {
    return -4;
  }

  return uECC_valid_point(_public, curve);
}

int uECC_compute_public_key(const uint8_t *private_key, uint8_t *public_key, uECC_Curve curve) {
  uECC_word_t _private[NUM_ECC_WORDS];
  uECC_word_t _public[NUM_ECC_WORDS * 2];

  uECC_vli_bytesToNative(_private, private_key, BITS_TO_BYTES(curve->num_n_bits));

  /* Make sure the private key is in the range [1, n-1]. */
  if (uECC_vli_isZero(_private, BITS_TO_WORDS(curve->num_n_bits))) {
    return 0;
  }

  if (uECC_vli_cmp(curve->n, _private, BITS_TO_WORDS(curve->num_n_bits)) != 1) {
    return 0;
  }

  /* Compute public key. */
  if (!EccPoint_compute_public_key(_public, _private, curve)) {
    return 0;
  }

  uECC_vli_nativeToBytes(public_key, curve->num_bytes, _public);
  uECC_vli_nativeToBytes(public_key + curve->num_bytes, curve->num_bytes, _public + curve->num_words);
  return 1;
}
